Polylogarithmic identities in cubical higher Chow groups

POLYLOGARITHMIC IDENTITIES IN CUBICAL
HIGHER CHOW GROUPS
Essen and Bayreuth, January, 1998, Herbert Gangl 1 and Stefan Müller-Stach 1
Abstract: Bloch asked whether it is possible to obtain the five term relation for the dilogarithm
in the context of the cubical version of his higher Chow groups, namely as the boundary of a 2dimensional
cycle in A 4 . We can give such a cycle and as a byproduct a realization of the isomorphism
B2(F )Q ∼ = CH 2 (F, 3)Q. We also shed some light on the relation between B3(F )Q and CH 3 (F, 5)Q by
proving further identities corresponding to functional equations of the trilogarithm, like the Kummer-
Spence relation. Finally, we exhibit distribution relations in CH m (F, 2m − 1)Q for each m.
I. Higher Chow Groups in a Nutshell
First let us give two different but equivalent definitions for Bloch’s higher Chow groups:
(1) Simplicial version (cf. [1]): Here ∆n = An F = affine n-space over an arbitrary field F with coordinates
t0, ..., tn, ti = 1 and Σ = ∪{ti = 0} ⊂ An is the union of all codimension one faces; a face is a
subsimplex ∆m ⊂ ∆n obtained by setting n − m coordinates equal to zero. Let Z p (X, n) be the free
abelian group of algebraic cycles of codimension p on X × ∆n , meeting all faces of all codimensions
properly. These groups form a simplicial abelian group:
...Z r →
→
(X, 3) Z
→
→
r →
(X, 2) → Z
→
r (X, 1) →
→ Zr (X, 0)
and CH p (X, n) is defined to be the n-th homotopy group of this simplicial group. In other words, via
the Dold-Kan theorem, CH p (X, n) is the n-th homology group of the complex
...Z p (X, n + 1) ∂ →Z p (X, n) ∂ →Z p (X, n − 1) ∂ →... ∂ →Z p (X, 0)
where ∂ = (−1) i ∂i is given by the alternating sum of restrictions to faces.
(2) Cubical version (cf. [10]): Here instead A n = (P 1 \ {1}) n and the faces are defined by xi = 0, ∞,
while the boundary is given by ∂ = (−1) i (∂ 0 i − ∂∞ i ). Zp (X, n) is defined as above, except that
one takes only the subcomplex of non-degenerate cycles. In [10] it is shown that this complex is
quasiisomorphic to the simplicial version.
Higher Chow groups satisfy several formal properties like the homotopy axiom and localization. They
also admit products and regulator maps to Deligne cohomology and étale cohomology. Their most important
property is given by the theorem of Bloch [1] (refined by Levine in [11]) which puts them into
relation with (the weight-graded pieces of) Quillen K-theory in the case of a smooth, quasiprojective
variety X of dimension d over a field F :
Let us mention a few further results:
Kn(X) (p) 1
⊗ Z[
(n + d − 1)! ] ∼ = CH p 1
(X, n) ⊗ Z[
(n + d − 1)! ]
Theorem([13],[16]). K M n (F ) ∼ = CH n (F, n).
1 Supported by the Deutsche Forschungsgemeinschaft
1

Theorem([14]). CH 2 (F, 3) = K ind
3 (F ) = K3(F )/K M 3 (F ).
Modulo torsion both sides are isomorphic to the Bloch group B2(F ) .
II. Computations in CH m (F, 2m − 1).
We work in cubical coordinates. Let G = Gn be the wreath product of the symmetric group
Sn and (Z/2Z) n . It acts on Z p (F, n) ⊗ Q via permutation and inversion of coordinates. Define
sgn : Sn → Z/2Z and sgn j : Z/2Z → Z/2Z to be the non-trivial 1-dimensional characters and
χ = sgn ·
then there is a natural choice of an idempotent
n
sgnj ,
j=1
Altn : Z p (F, n) ⊗ Q → Z p (F, n) ⊗ Q , Z ↦→ 1
|G|
χ(g) g(Z) ,
Note that we abbreviate Altn and sometimes Alt (when the context is clear) for the alternation over
the full group G. Define C p (F, n) ⊂ Z p (F, n) ⊗ Q to be the image of Alt together with the differential
induced by ∂. The resulting homological complex
C m (F, ·) : . . . → C m (F, 2m) → C m (F, 2m − 1) → . . . → C m (F, m) → 0
still computes CH m (F, n)Q by [10].
I. First let m = 2: the complex C 2 (F, ·) has the acyclic subcomplex
. . . → C 1 (F, 1) ∧ C 1 (F, 3) → C 1 (F, 1) ∧ C 1 (F, 2) → C 1 (F, 1) ∧ ∂C 1 (F, 2) → 0 ,
i.e. the truncation of the subcomplex consisting of subvarieties where one coordinate entry is constant.
The proof of acyclicity is the same as in [12]. The quotient complex will be denoted by A 2 (F, ·). It
is quasiisomorphic to C 2 (F, ·) and has certain advantages; for example cycles in C 2 (F, 3) with one
coordinate entry being constant have zero image in A 2 (F, 3). We call such cycles negligible. Another
advantage is—due to the fact that we mod out C 2 (F, 2) by C 1 (F, 1) ∧ ∂C 1 (F, 2)—that the following
diagram is well-defined and commutative:
Z[F − {0, 1}]
ρ2
A2 (F, 3)
∂A2 (F, 4)
β2
∂
2 F ∗
g∈G
id⊗id
2 A (F, 2)
where for a group G we denote by 2 G the subgroup of G ⊗ G generated by {a ⊗ b − b ⊗ a | a, b ∈ G}.
The map β2 is given on generators as [a] ↦→ a ⊗ (1 − a) − (1 − a) ⊗ a.
The map ρ2 is defined as ρ2(a) = Ca mod ∂A 2 (F, 4) for a ∈ F ∗ , where Ca is defined in (2) below.
Notation: Given a rational map ϕ : (P 1 ) d → (P 1 ) n , we let Nϕ be the cycle associated to ϕ,
Nϕ := ϕ∗
(P 1 ) d
∩ n ,
2
(1)

i.e. the direct image in the sense of Fulton (1.4) [6].
For mnemonic reasons (cube=[ ]) we propose a “cubical” notation
[ϕ1(x), . . . , ϕn(x)] , x = (x1, . . . , xn),
for Altn(N (ϕ1(x),...,ϕn(x))).
By abuse of notation, we will often identify ϕ with Nϕ , and we introduce the further abbreviation for
parametrized cycles of the special form associated to a rational function f
Zf = Alt3 N(x,1−x,f(x)) .
Finally, we return to the definition of Ca (cf. [16]) as
Ca = Z x−a
x
= [x, 1 − x,
x − a
] . (2)
x
Note: In the following, equality signs between cycles are understood modulo negligible terms.
Lemma 1: Let f, g, r and s be rational functions in one variable. Then there exists an element
W ∈ A2 (F, 4) with
∂W = f(x), g(x), r(x)s(x) − [f(x), g(x), r(x)] − [f(x), g(x), s(x)] .
if all three cycles on the right hand side are admissible, i.e. intersect all faces in the correct dimension.
Proof. Let W be the parametrized cycle W := [f(x), g(x), y−r(x)s(x)
y−r(x) , y]. Then we compute (denoting
the coordinates by x1, . . . , x4) that W ∩ {x1 = 0}, W ∩ {x1 = ∞}, W ∩ {x2 = 0} and W ∩ {x2 = ∞}
are negligible, since one coordinate entry becomes constant. Next, W ∩ {x4 = ∞} = ∅, since the third
coordinate becomes 1. The remaining three boundaries give the three terms in the assertion.
Corollary 1. (a) Ca + C1−a = C1 in A 2 (F, 3)/∂A 2 (F, 4).
(b) Consider, for arbitrary e, f ∈ F × , the cycle W (a, b, c, d) : x ↦→ [ x−a x−b x−c
e , f , x−d ] for pairwise
distinct elements a, b, c, d ∈ F . Then
W (a, b, c, d) = C c−a
b−a
− C d−a ∈ A
b−a
2 (F, 3)/∂A 2 (F, 4)
Furthermore,
[x, 1 − x a
, 1 −
b x ] = C a
b ∈ A2 (F, 3)/∂A 2 (F, 4) .
Proof: (a) We apply Lemma 1 with f(x) = x, g(x) = 1 − x, r(x) = x−a
x−1
x−1 and s(x) = x to obtain
Ca = Z x−a
x
yields
= Z x−a
x−1
+ C1. Substituting x ↦→ (1 − x) and permuting the first two components in Z x−a
x−1
x − a
1 − x − a
x − (1 − a)
x, 1 − x, = 1 − x, x, = − x, 1 − x, = −C1−a .
x − 1
−x
x
(b) We only prove the first assertion (the second can be shown in a similar way), and we restrict to
the case e = f = 1. The general case can be easily deduced by repeating some of the arguments used.
W (a, b, c, d) = W (0, b − a, c − a, d − a) by the transformation x ↦→ x + a. However
W (0, b − a, c − a, d − a) = [x, 1 − x x + a − c
,
b − a x + a − d ]
1 x+a−c
(here we multiplied with the admissible cycle [x, a−b , x+a−d ] ∈ C1 (F, 1) ∧ C1 (F, 2), which is allowed
by lemma 1 in view of the definition of A 2 (F, ·)) and therefore, using the transformation x ↦→ (b − a)x,
W (0, b − a, c − a, d − a) = [(b − a)x, 1 − x,
3
c−a x − b−a
x − d−a
b−a
] ,

whence it ensues that it is equal (up to an admissible cycle in C 1 (F, 1) ∧ C1 (F, 2)) to
c−a
c−a
x − b−a
x − b−a
[x, 1 − x, ] = [x, 1 − x, ] − [x, 1 − x,
x
x − d−a
b−a
d−a x − b−a
] = C c−a − C d−a .
x
b−a b−a
We proceed to describe the geometric nature of certain functional equations for the dilogarithm in the
realm of higher Chow groups.
Theorem 1: (a) If F contains a primitive n-th root of unity, then every a ∈ F ∗ gives rise to a
distribution relation:
nCan = n2
ζn Cζa in A
=1
2 (F, 3)/∂A 2 (F, 4)
(b) For a = b, 1 − b, and a, b = 0, 1, one obtains the five term relation
Va,b := C a(1−b) − C 1−b + C1−b − C a
b(1−a) 1−a
b + Ca = 0 ∈ A 2 (F, 3)/∂A 2 (F, 4) .
(c) The inversion relation holds:
2(Ca + C 1
a − 2C1) = 0 ∈ A 2 (F, 3)/∂A 2 (F, 4) .
(d) Let f : P1 → P1 be any rational function of degree n. Then the following relation holds:
2n 2 Ccr(f(x),a,b,c) = 2n
Ccr(x,α,β,γ) γ∈f −1 (c) β∈f −1 (b) α∈f −1 (a)
assuming that x, a, b, c ∈ F and all α, β, γ are mutually distinct and lie in F .
Here cr(a, b, c, d) denotes the cross ratio (a−c)(b−d)
(a−d)(b−c) .
Proof: (a) We give the proof for n = 2: note that 2C a 2 = [t 2 , 1 − t 2 , t2 −a 2
t2 ] (cf. [6], 1.4). Therefore
by repeated application of Lemma 1 (check that in each step all cycles are defined, i.e. admissible):
2Ca2 = [t2 , 1 − t2 , t2−a 2
t2 ] = [t2 , 1 − t2 , t−a
t ] + [t2 , 1 − t2 , t+a
t ]
= [t2 , 1 − t, t−a
t ] + [t2 , 1 + t, t−a
t ] + [t2 , 1 − t, t+a
t ] + [t2 , 1 + t, t+a
= [t2 , 1 − t, t−a
t ] + [t2 , 1 − t, t+a
t ] + [t2 , 1 − t, t+a
t ] + [t2 , 1 − t, t−a
= 4[t, 1 − t, t−a
t ] + 4[t, 1 − t, t+a
t ]
= 4Ca + 4C−a .
For n ≥ 3, the proof is similar, noting that nCan = [tn , 1 − tn , tn−a n
tn t n − a n = (t − ζa).
t ]
t ]
] and 1 − t n = (1 − ζt) and
(b) In table 1 a proof of the five term relation is given both in terms of cycles and (parallel to it)
in terms of graphs; the latter visualize (and guide) the decomposition of cycles. One can play the
following game: starting from a “distinguished position” (this corresponds to a sum of cycles Cai),
one is allowed to perform a number of “moves” (this corresponds to decomposing the cycles) of a
certain kind which should eventually lead to another “distinguished position” (again, a sum Cbj ).
If the latter is different from the starting position, the difference [ai] − [bj] gives a functional
equation for the dilogarithm. The (undirected) graph encodes certain data of cycles whose coordinates
are products of fractional linear transformations: we fix an ordering of the coordinates; the zeros of a
coordinate are marked points which are connected via marked edges to its poles—also given as marked
points. The mark underneath a point denotes the point viewed as lying in P 1 , the encircled number
above an edge gives the number of the coordinate in the chosen ordering of the cycle coordinates.
E.g. if the second coordinate is given by 1 − x, it is encoded as
4

2
• • .
1 ∞
The other information which is captured in the graph are the preimages of 1, these are pictured by
a full-square with both the marking of the point (below) and the information about the coordinate
number inside the square. In the above example we get 0 for the second coordinate, and this results
in the picture 2 .
0
A dotted square in the example above means that none of the coordinates involved has image 1
at this point.
Fractional linear cycles are admissible if and only if in their corresponding graph any two adjacent
edges are “glued” along a full-squared vertex (as opposed to a dotted-squared one). The latter condition
corresponds in the language of cycles to boundary terms which have one coordinate being 0 or
∞ (and are a priori not admissible) but which are annihilated since another coordinate is equal to 1
and therefore vanishes.
A distinguished position is of the type • , the information about coordinates
is essentially redundant (but convenient for purposes of comparison as in the table).
A move is one of the following: (1) split up an edge into two, (2) move a full square from one vertex
to another, and (3) the reverse operations to (1) and (2).
A move is admissible if both graphs before and after the move are admissible.
We could also introduce a direction of the graph, say, by substituting edges by arrows going from
a zero to a pole and then we can give it a sign, depending on the permutation of the encircled marks
and the direction of the arrows. This will determine a cycle unambiguously from its graph. Since we
can get by without and we don’t want to overload the picture, we omit it.
(c) follows from (b) and of corollary 1, (a):
Va,b + V 1 1
a , b − (Cb + C1−b − C1) − (C 1
b + C1− 1
b − C1) = Ca + C 1
a − Cb − C 1
b − (C a + C b − 2C1) .
b a
Now add the corresponding expression where the roles of a and b are interchanged.
(d) For simplicity we restrict to the case of f(x) being a polynomial of degree n and taking a, b, c ∈ F
such that all their preimages are distinct (i.e. they are non-critical points of the map f). Then for
any α and β such that f(α) = a, f(β) = b, the following decompositions hold:
f(x) − a
b − a =
i (x − αi)
,
(β − αi)
i
f(x) − b
a − b =
(x − βj)
j
j
(α − βj)
Therefore reparametrizing nCc, using x ↦→ f(x)−a
b−a , gives
i
and
(x − αi) f(x) − b
nCc =
,
(β − αi) a − b ,
k (x − γk)
(x − αl)
and we can decompose in the first coordinate in the following way
=
x − αi
,
β − αi
f(x) − b
a − b ,
k (x − γk)
(x − αl)
i
l
l
f(x) − c
f(x) − a =
k (x − γk)
.
(x − αl)
since all the resulting cycles are admissible, and furthermore, we can decompose in the last coordinate
which for convenience we write as a double product (which is matched up by a factor n on the left)
n 2 Cc =
x − αi
,
β − αi
f(x) − b x −
γk
, .
a − b x − αl
i,k,l
6
l

Now we need to distinguish two cases i = l and i = l. The latter one poses no problem, we can
, and a typical cycle
decompose in the second coordinate into terms x−βj
α−βj
x − αi
,
β − αi
x − βj
,
α − βj
x − γk
x − αl
is admissible and—via corollary 1(b)—equivalent to C γ k −α i
β j −α i
− C αl−αi .
βj −αi For the terms with i = l we only need to decompose the second coordinate into factors x−βj
αi−βj with
−1 the same αi above we could have chosen any α ∈ f (a) to make sure of the admissibility of the
. This gives us the
resulting cycles, and then we can again invoke corollary 1(b) to obtain C γ k −α i
β j −α i
claim with a polynomial f, since two terms t(i, l) = C α l −α i
β j −α i
that 2(Cx + C x
x−1
and t(l, i) are equal up to 2-torsion (note
) = 0, which follows from (c) and corollary 1(a)).
Corollary 2. C1 = C−1 = 0 ∈ CH 2 (F, 3)Q.
Proof: Use (a) with n = 2 and (c) with a = −1, respectively. The outcome is 4C−1 = 4C1 and
4C−1 = −2C1, yielding 6C1 = 12C−1 = 0.
Returning to diagram (1), we see that the map induced by ρ2 on the kernels factors (up to tensoring
with Q) through the Bloch group B2(F ) (which is defined [14] as the quotient of ker β2 by
the subgroup generated by the five term relation), thereby realising the well-known isomorphism
B2(F )Q ∼ = CH 2 (F, 3)Q.
II. Now let m ≥ 3. Consider the cycles of Bloch-Kriz ([2])
Ca = C (m)
a : (x1, ..., xm−1) ↦→ x1, ..., xm−1, 1 − x1, 1 − x2
, ..., 1 −
x1
xm−1
, 1 −
xm−2
a
xm−1
in C m (F, 2m − 1) (m will be clear from the context so we suppress the superscript).
As in the case m = 2 we modify the complex C m (F, ·) in such a way that we can manipulate
parametrized cycles modulo negligible ones. To simplify our computations a little bit, we mod out by
a subcomplex J m (F, ·) which was already considered by Bloch and Kriz in [2]. It can be constructed
as follows: look first at the graded algebra
C p (F, n)
p,n
with multiplication given by the product in cubical coordinates (see [11]). The differential graded
ideal generated by all C p (F, n) with n ≥ 2p + 1 and all C p (F, 2p) for p ≥ 1 is denoted by J · (F, ·). It
contains for example all groups ∂C p (F, 2p) for p ≥ 1.
For m = 3, the group J 3 (F, 5) contains the subgroups
J 3 (F, 4) contains the subgroups
C 1 (F, 1) ∧ C 2 (F, 4), C 1 (F, 2) ∧ C 2 (F, 3) and ∂C 3 (F, 6).
∂C 1 (F, 2) ∧ C 2 (F, 3), C 1 (F, 1) ∧ ∂C 2 (F, 4) and C 1 (F, 2) ∧ C 2 (F, 2).
7

The quotient complex C 3 (F, ·)/J 3 (F, ·) will be denoted by A 3 (F, ·). For number fields F , the complex
J 3 (F, ·) is known to be acyclic, so A 3 (F, ·) is quasiisomorphic to C 3 (F, ·) in this case. This is a
consequence of Borel’s theorem [4] by using the formulas for the regulator in [8].
In general, the acyclicity of J 3 (F, ·) is not known and is related to the Beilinson-Soulé conjecture for
m = 2 (cf. [2] and [15]).
In the following we will frequently omit terms which are contained in J 3 (F, ·) and call them—as
the corresponding ones in the case m = 2—negligible. As before, equality signs between cycles are
understood modulo negligible terms.
We have again a commutative diagram
Z[F − {0, 1}]
ρ3
A3 (F, 5)
∂A3 (F, 6)
β3
∂
Z[F − {0, 1}]
R2(F )
ρ2∧id
3
A (F, 4)
Here R2(F ) is defined as the subgroup generated by five term relations.
The map β3 is defined on generators as [a] ↦→ [a] mod R2(F ) ∧ a.
The map ρ3 is defined as [a] ↦→ Ca = C (3)
a .
The next lemma tells us how far the decomposition of certain cycles is from being bounded by some
admissible cycle.
Lemma 4: Consider a parametrized cycle of the form
[f1(x), f2(y), f3(x), f4(x, y), f5(y)]
where all fi are rational functions and f4 is a product of fractional linear transformations when
considered as a function in the variable y.
(a) Assume that we can write f1(x) = g(x)h(x) for some rational functions g and h. Then the
admissible cycle
W := [
∧ F ∗
z − g(x)h(x)
, z, f2(y), f3(x), f4(x, y), f5(y)] ∈ A
z − g(x)
3 (F, 6)
gives the following relation (provided all terms are admissible):
∂(W ) = [f1(x), f2, f3, f4, f5] − [g(x), f2, f3, f4, f5] − [h(x), f2, f3, f4, f5]
+
div(f4)
±[
z − g(x)h(x)
, z, f2(y(x)), f3(x), f5(y(x))],
z − g(x)
where we solve the equations f4(x, y) = 0 (coefficient is +1) and 1/f4(x, y) = 0 (coefficient is −1) by
the implicit function y(x) in terms of x. Specifically, if e.g. f2(y(x)) = f3(x), the corresponding term
in the latter sum vanishes under the alternation.
A similar relation holds if we decompose fi for i = 4 into two factors.
(b) In the case i = 4, assume that we can write f4(x, y) = g(x, y)h(x, y) for some rational functions g
and h. Then there is an admissible cycle W ∈ A 3 (F, 6) such that
∂(W ) = [f1, f2, f3, f4(x, y), f5] − [f1, f2, f3, g(x, y), f5] − [f1, f2, f3, h(x, y), f5] ,
8
(3)

provided that all occurring terms are admissible.
(c) Assume for each solution y = r(x) of f4(x, y) = 0 that f1(x) = f2(r(x)) = g(x)h(x) and that
either g(r(x)) = g(x) or g(r(x)) = h(x).
Introduce a shorthand Z(e1, e2) = [e1(x), e2(y), f3(x), f4(x, y), f5(y)] for any two functions e1, e2 in
one variable. Then
2Z(f1, f2) = Z
g, f2 + Z h, f2 + Z f1, g + Z f1, h
and
2Z(f1, f2) = 2 Z(g, g) + Z(g, h) + Z(h, g) + Z(h, h) .
Proof. (a) The first three boundaries W ∩ {x1 = 0} , W ∩ {x1 = ∞} and W ∩ {x2 = 0} produce
the terms [f1(x), f2, f3, f4, f5] − [g(x), f2, f3, f4, f5] − [h(x), f2, f3, f4, f5]. Then W ∩ {x2 = ∞} = ∅,
since the first coordinate becomes 1. The next four boundaries W ∩ {x3 = 0} , W ∩ {x3 = ∞},
W ∩{x4 = 0} and W ∩{x4 = ∞} are in J 3 (F, 5) and therefore negligible. The boundaries W ∩{x5 = 0}
and W ∩ {x5 = ∞} create the sum
div(f4)
boundaries W ∩ {x6 = 0} and W ∩ {x6 = ∞} are again negligible.
(b) is easier to prove than (a) and therefore left to the reader.
(c) The first equation follows by using the boundaries of the cycle
±[ z−g(x)h(x)
z−g(x) , z, f2(y(x)), f3(x), f5(y(x))]. The last two
[ z − f1(x)
z − g(x) , z, f2(y),
z − f2(y)
f3(x), f4(x, y), f5(y)] − [f1(x),
z − g(y) , z, f3(x), f4(x, y), f5(y)] ,
and noting that the “deviation terms” (as given in (a)) from the decomposition are the same for both
summands up to alternation. So they cancel in the difference and we are left with
Z
g, f2 + Z h, f2 + Z f1, g + Z f1, h − 2Z(f1, f2) = 0 .
For the second equation, one uses the first one and the boundaries of the cycles
[ z − f1(x)
z − g(x) , z, g(y), f3(x),
z − f1(x)
f4(x, y), f5(y)] + [
z − g(x) , z, h(y), f3(x), f4(x, y), f5(y)] and
[g(x),
z − f2(y)
z − g(y) , z, f3(x),
z − f2(y)
f4(x, y), f5(y)] + [h(x),
z − g(y) , z, f3(x), f4(x, y), f5(y)] .
where again all the “deviation terms” cancel under the alternation.
Theorem 2: (a) The Kummer-Spence relation for the trilogarithm
S(a, b) := C b(1−b) + C a(1−b) + C (1−a)(1−b) − 2C b − 2C 1−b − 2C b − 2C 1−b + 2C 1 + 2C 1
a(1−a) b(1−a)
ab
a
a
1−a
1−a
1−a a − 2C1− 1
b
holds in the form 4 S(a, b) = 0 in A 3 (F, 5)/∂A 3 (F, 6) whenever a = b, 1 − b, and a, b = 0, 1.
(b) The inversion relation and the “3-term relation” hold
4(Ca − C 1
a ) = 0 4(Ta − Tb) = 0 , for Ta = Ca + C 1
1−a + C 1− 1
a .
Proof: (a) All cycles involved will be admissible.
If we use a phrase like “decompose the ith coordinate uv of a cycle into u and v”, it is understood
that there is an admissible cycle whose boundary is just the difference of the given cycle and the (sum
of the) resulting ones after formally replacing the ith coordinate by u and v.
9

Let ϕ1(x) = x(1−x)
a(1−a) , ϕ2(x) = a(1−x)
x(1−a) and ϕ3(x) = (1−a)(1−x)
ax . Reparametrizing Cϕ1(b) via x ↦→ ϕ1(x),
y ↦→ ϕ1(y) produces the factor (deg ϕ1) 2 = 4 in the following equality
4Cϕ1(b) = [ϕ1(x), ϕ1(y), 1 − ϕ1(x), 1 − ϕ1(y) ϕ1(b)
, 1 − ] .
ϕ1(x) ϕ1(y)
We apply lemma 4(c) twice to this cycle, first with u(x) = x(1−x), thereby getting rid of the constants
a(1 − a) in the first two coordinates, and then with u(x) = x, yielding four cycles all of which are the
same after reparametrization, namely
[x, y, (1 − x x
)(1 −
a 1 − a
y y
), (1 − )(1 −
x 1 − x
b b
), (1 − )(1 − )] .
y 1 − y
Using lemma 4(b), we decompose the fourth coordinate into 1− y
y
x and 1− 1−x , leaving us with Z1+Z2,
where
Z1 = [x, y, (1 − x x y b b
a )(1 − 1−a ), 1 − x , (1 − y )(1 − 1−y )] and
Z2 = [x, y, (1 − x x
y b b
a )(1 − 1−a ), 1 − 1−x , (1 − y )(1 − 1−y )] .
Since we had four such cycles, we obtain 4C ϕ1(b) = 4(Z1 + Z2).
We will now construct Z2 in a different way, using ϕ2(b) and ϕ3(b), and then subtract the outcome
from the one above. Consider
C ϕ2(b) = a(1 − x)
x(1 − a)
a(1 − y) x − a y − x
, , ,
y(1 − a) x(1 − a) y(1 − x) ,
b − y
b(1 − y)
and use, like above, lemma 4(c) with u = a/(1 − a) to get rid of the constants in the first two
coordinates. Then we apply lemma 4(a) to decompose the third coordinate into x−a 1
1−a and x . Note
that the cycle with 1
x is just C1−b−1 (to be precise, we need lemma 4(c) again with u = −1).
The same procedure applies to Cϕ3(b), the only difference is that the role of a and 1 − a in the third
coordinate are interchanged.
We again invoke lemma 4(a), noting that f1(x) = f2(y(x)) for y(x) = x, and we can “merge” the two
remaining cycles in the third coordinate to
1 − x
x
1 − y
, ,
y
x − a
1 − a
x − 1 + a
· ,
a
y − x
Successive decomposition of the fourth coordinate into y−x
1−x
then of the fifth coordinate into b−y 1
1−y and b
one non-negligible cycle
1 − x
x
1 − y
, ,
y
x − a
1 − a
y(1 − x) ,
and 1
y
b − y
.
b(1 − y)
(the second one is negligible), and
(again, the second one is negligible) leaves us with only
x − 1 + a
· ,
a
y − x
1 − x
b − y
, ,
1 − y
which now finally can be decomposed in the first two coordinates via lemma 4(c) with u(x) = 1 − x
into four terms Ti given (after an obvious reparametrization) by
= x, y, (1 − x
1 − a
− x, y, (1 − x
1 − a
)(1 − x
a
), 1 − y
x
x y
)(1 − ), 1 −
a 1 − x
T1 + T2 − T3 − T4 :=
Let us decompose both T2 in the fourth coordinate into 1 − y
x
1 − b
x x y x y − b
, 1 − + x, y, (1 − )(1 − ), (1 − ) ,
y
1 − a a x x − 1 y − 1
1 − b
x x y − 1 y − b
, 1 − − x, y, (1 − )(1 − ), (1 − )x , .
y
1 − a a 1 − x x y − 1
(this will give a negligible cycle
in C1 (F, 2) ∧ C2 (F, 3)) and then T4, also in the fourth coordinate, into 1 − y
and x
x−1
and call the resulting non-negligible terms T ′ 2 and T ′ 4, respectively.
10
1−x
and x
x−1 (negligible),

T1 and T ′ 2 can be merged to Z1, T3 and T ′ 4 can be merged to Z2. So we have constructed Z2 in a
different way, as promised.
Summarizing, we can write C ϕ2(b) + C ϕ3(b) = Z1 − Z2 + 2C 1−b −1, and with the above,
3
Cϕj(b) = 2Z1 + 2C1−b−1 .
j=1
at least if we multiply both expressions by 4, which gives the desired result using that Z1 can be
decomposed—by iterated application of lemma 4(a)—into cycles of the form Cα:
Z1 = C 1−b + C 1−b + C b + C b − C 1 − C 1
1−a
a
1−a a 1−a a .
b−1
(b) Let us introduce a shorthand 〈x〉 = Cx − C 1 , and B =
x b
S(a, b) − S(a, 1 − b) =
a(1 − b)
+
b(1 − a)
S(b, a) − S(b, 1 − a) =
Adding them up, we conclude
(1 − a)(1 − b)
ab
b(1 − a)
+
a(1 − b)
a−1
and A = a , then
b
+ 2 = 〈B/A〉 + 〈BA〉 + 2〈1/B〉 and
b − 1
(1 − b)(1 − a)
ba
2〈AB〉 − 2〈A〉 − 2〈B〉 = 0 .
a
+ 2 .
a − 1
Rewriting this gives 2〈B/A〉 + 2〈BA〉 − 2〈B 2 〉 = 0, and together with the above and the distribution
relation for n = 2 we obtain 4〈−B〉 = 0.
Similarly, we consider
2
b(1 − b)
a(1 − b)
S(a, b) − S(b, a)) = 2
+
a(1 − a) b(1 − a)
+ 4Ta − 4Tb
(we have left out terms of the form 4〈x〉 in the first equation), and using that the term in brackets on
the right is equal to 〈 1−b
1−a 〉 by the above, we are done.
For the record, we mention also the validity of the distribution relations in general.
Proposition 8: If F contains a primitive n-th root of unity, then every a ∈ F gives rise to a
distribution relation for any m:
n m−1
Can = n2m−2 Cζa in A m (F, 2m − 1)/∂A m (F, 2m) .
ζ n =1
Proof: Due to the special form of the cycle
x n 1 , . . . , x n m−1, 1 − x n 1 , 1 − xn2 xn , . . . , 1 −
1
an
xn
,
m−1
it can be shown that one can decompose the last m coordinates modulo ∂A m (F, 2m) into linear factors:
the additional boundary terms which occur in lemma 4 do not appear since, for a typical solution
xi = ζnxi+1 of the corresponding equation, the resulting cycle vanishes under the alternation.
These decompositions are already sufficient to deduce the distribution relations along the lines of
proposition 2 (a).
11

Concluding Remarks: Our computations for m = 3 could be important to study the relation
between the third Bloch group B3(F ) = ker β3/R3(F ) and CH 3 (F, 5). Here R3(F ) is defined as the
subgroup of Z[P1 F −{0, 1, ∞}] generated by Goncharov’s relation, the 3-term relation and the inversion
relation (cf. [9], (5.17)).
Namely, if F is a field which has the property that CH 3 lin (F, 5)Q (in the sense of [7]) surjects onto
CH 3 (F, 5)Q, then we conjecture that also the map ρ3 : B3(F )Q → CH 3 (F, 5)Q is surjective. This can
probably be proved with our method of decomposing and merging fractional linear cycles.
If F is a field such that the map CH 3 lin (F, 5)Q → CH 3 (F, 5)Q is even an isomorphism, then one can
construct a well-defined inverse map to ρ3 which is, on generators, defined by
1 a1x + b1 1y + c11 a1 2x + b12 y + c1 , . . . ,
2
a51 x + b51 y + c51 a5 2x + b5 2y + c5 2
↦→ (−1)
2 i1,...,i5=1
i1+...+i5
⎛
a
⎝
1 a i1
, . . . ,
5 i5
b5 i5
c5 0
, 0
1
i5
⎞
⎠
as a linear combination of cross ratios of 6 points in P 2 F
b 1 i1
c 1 i1
(cf. [8]) and thus giving rise to an element in
B3(F ). Therefore, if the map CH 3 lin (F, 5)Q → CH 3 (F, 5)Q is bijective, then CH 3 (F, 5)Q ∼ = K (3)
5 (F )Q
has a presentation in terms of generators and relations as conjectured by Zagier in the case of a number
field.
For number fields a cubical version of the result of [7], together with the solution of the rank conjecture
([5]), imply that the map of Gerdes is at least surjective. Hence a more careful study of this map in
terms of group homology has to be done in order to prove Zagier’s conjecture.
Acknowledgements: We are indebted to Spencer Bloch for posing the original problem, and it
is a pleasure to thank him and Masaki Hanamura for stimulating discussions and Rob de Jeu for
pointing out a serious gap in an earlier version of the paper.
We would also like to thank the organizers of the Seattle meeting for the opportunity to present our
work. The AMS and DFG have offered financial support. The first author expresses his gratitude
to the Institute for Experimental Mathematics, Essen, and the Max-Planck-Institut für Mathematik,
Bonn, for their hospitality.
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